https://file.aiquickdraw.com/mathbot/file/1726892270892lzoslhx4.jpg

Let's analyze the problem step by step.

### Given:
- The box has a **height of 60 inches**.
- The **length is 2.5 times the width** (we'll call the width \( x \)).
- The **perimeter of the base** is the sum of the lengths of the two dimensions: length and width, calculated as \( 2(x + l) \), where \( l \) is the length.
- The total sum of the perimeter of the base and the height cannot exceed **130 inches**.

### Step 1: Express the perimeter and length.
- The length \( l \) is given as \( l = 2.5x \).
- The perimeter of the base of the box is \( 2(x + l) = 2(x + 2.5x) = 2(3.5x) = 7x \).

### Step 2: Create the inequality.
We are told that the **sum of the perimeter** of the base and the **height** cannot exceed 130 inches, so we set up the inequality:
\[
7x + 60 \leq 130
\]

### Step 3: Solve the inequality.
\[
7x + 60 \leq 130
\]
Subtract 60 from both sides:
\[
7x \leq 70
\]
Now divide by 7:
\[
x \leq 10
\]

Thus, the inequality for \( x \) is:
\[
0 < x \leq 10
\]

### Conclusion:
The correct answer is **A) \( 0 < x \leq 10 \)**.

Let me know if you would like more details on any step. Here are some related questions for further practice:

1. If the height of the box were 50 inches instead of 60 inches, what would be the maximum allowable width?
2. How would the inequality change if the length was 3 times the width instead of 2.5 times the width?
3. What is the minimum perimeter of the base if the width is 10 inches?
4. What would happen if the shipping service allowed the sum of the perimeter and height to be 140 inches?
5. How would you calculate the volume of the box given these dimensions?

**Tip:** Always break down word problems into smaller parts and represent unknown quantities with variables for clarity.

A shipping service restricts the dimensions of the boxes it will ship for a certain type of service. The restriction states that for boxes shaped like rectangular prisms, the sum of the perimeter of the base of the box and the height of the box cannot exceed 130 inches. The perimeter of the base is determined using the width and length of the box. If a box has a height of 60 inches and its length is 2.5 times the width, which inequality shows the allowable width x, in inches, of the box?

Learn how to solve a real-world shipping box problem involving the sum of perimeter and height restrictions. This problem uses algebraic inequalities to determine the maximum allowable width of the box when the length is 2.5 times the width and the height is 60 inches.

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